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Results tagged with big-list
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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
11
votes
Different ways of thinking about the derivative
There are two closely related interpretations of the derivative of a generating function in combinatorics.
If $A(x) = \sum a_n x^n$ counts the number of $A$-structures on an $n$-element ordered set, …
9
votes
Fixed point theorems
Here is a teeny tiny toy version of the Lefschetz fixed point theorem: let $f : S \to S$ be an endomorphism of a finite set and let $K[f] : K[S] \to K[S]$ be the induced linear map on free vector spac …
5
votes
Problems where we can't make a canonical choice, solved by looking at all choices at once
Here's a basic but important example. In the modern approach to things like algebra we study objects like groups and rings in two stages: first we study their abstract structure, then we study their …
7
votes
"Surprising" categorical equivalences
The tangle category $\text{Tang}$ is the category whose objects are the non-negative integers $n$ and whose morphisms $n \to m$ are collections of paths from $n$ points arranged on a line in $\mathbb{ …
7
votes
"Surprising" categorical equivalences
The $2$-cobordism category $2\text{Cob}$ is the category whose objects are (compact, oriented) $1$-manifolds and whose morphisms $M \to N$ are (compact, oriented) $2$-manifolds with boundary $M \sqcup …
10
votes
"Surprising" categorical equivalences
The forgetful functor from topological abelian monoids to topological spaces has a left adjoint, the infinite symmetric product $\text{SP}$. The Dold-Thom theorem asserts that if $X$ is a CW-complex, …
1
vote
Random versions of deterministic problems
This may not be quite what you had in mind, but: suppose you were trying to compute the absolute value of a Gauss sum $\sum_{a=0}^{p-1} \zeta^{a^2}$ where $\zeta = e^{ \frac{2 \pi i}{p} }, p$ an odd p …
12
votes
Undergraduate Level Math Books
The Princeton Lectures in Analysis by Stein and Shakarchi are great introductions to Fourier, complex, and real analysis (in that order!).
26
votes
Looking for book with good general overview of math and its various branches
I'm going to answer your title question instead of your body question (which to my mind is completely different): what you're looking for in the title is the Princeton Companion to Mathematics.
18
votes
Particular problem solved by solving a more general problem.
Frequently in mathematics the best way to determine the value of a sequence at a particular index is to compute its value at every index, even though the latter seems on the surface like a harder prob …
12
votes
Theorems with unexpected conclusions
Here's one I was reminded of recently. Recall that a projective plane is a triple $(P, L, I)$ where $P$ is a set of "points," $L$ is a set of "lines," and $I$ is a subset of $P \times L$ describing t …
17
votes
Most harmful heuristic?
Also not really a heuristic, but "differentiation is easy," as encoded in the following two sub-heuristics:
Differentiation is just repeated application of the product and chain rules, and
Most func …
28
votes
Applications of mathematics
A particularly striking application to physics and chemistry is explained in Singer's book Linearity, symmetry, and prediction in the hydrogen atom. The practical problem, in the large, is easy to sta …
20
votes
What are some examples of colorful language in serious mathematics papers?
I always liked Edward Burger's A Tail of Two Palindromes. It begins as follows:
Upon a preliminary perusal, this parable may appear to be about pairs of palindromes, periods, and pitiful alliteration …
52
votes
Theorems with unexpected conclusions
I learned this example from Noam Elkies's excellent article The Klein Quartic in Number Theory. Elkies observes that Siegel's 1968 paper Zum Beweise des Starkschen Satzes, in order to prove its main …